Desai and Lieber 



body of information on the flows as they exist in nature, it is evident that what- 

 ever information we seek to provide must not contradict experience; indeed, it 

 must be taken from experience. Then we can be reasonably assured of its con- 

 sistency with the implicit information presumed to be embodied in the Navier- 

 Stokes equations. ^ 



A natural question which now arises is this: What is the common factor in 

 all our experimental and theoretical experience ? The answer seems to be the 

 prominent place of potential flows. All real flows are always found to have 

 some parts of their flow field potential under suitable conditions and, more sig- 

 nificantly, they can be made increasingly potential by definite manipulations of 

 the flow parameters. Impulsively started motions are always potential in the 

 initial stages. All existing theories, e.g., the boundary layer theory; water 

 waves theory; wing theory; and the subsonic, supersonic, and hypersonic flow 

 theories etc., give a central place to the potential flows. This realization leads 

 us to believe that the key information can be provided by the potential flows. 



The next question to arise is: How shall we use the available explicit in- 

 formation about potential flows in the Navier -Stokes equations to make explicit 

 the information about flows as observed in nature without imposing any restric- 

 tions on the flow parameters? It has been our conviction that one way to realize 

 the evolution of a flow field characterized by nonlinear equations is through a 

 process of iterations which yields a linear set of equations and that this process 

 does not need to be assumed a priori to be valid for any specific range of the 

 characteristic parameters that are not for any particular part of the flow do- 

 main. In fact, we think that this process is of fundamental importance in arriv- 

 ing at a mathematical solution of nonlinear partial differential equations, such 

 as the Navier -Stokes equations, which contain information about physical phe- 

 nomena. 



It is a remarkable fact that a very few cases exist for which explicit infor- 

 mation bearing on the motion of actual flows has been extracted from the Navier- 

 Stokes equations by strictly mathematical procedures, i.e., without using explicit 

 information obtained from extramathematical sources. In the context of this 

 work, it is equally relevant to note and emphasize that when explicit analytical 

 and experimentally verifiable information has been derived from the Navier- 

 Stokes equations, in almost all such cases explicit information derived from 

 extramathematical sources was employed. Such extramathematical information 

 is usually introduced by making judicious simplifying assumptions based on the 

 observation of phenomena and the critical examination of experiments for which 

 they are appropriate. Accordingly, a good simplifying assumption is a way of 

 explicitly stating information which is already implicitly contained in the Navier - 

 Stokes equations, in such cases for which the simplifying assumption is valid. 

 Indeed, we may regard this as a mathematical criterion for a simplifying as- 

 sumption to be appropriate in a particular case. That is, if we could solve the 

 Navier -Stokes equations for such a specific case, the solution obtained would 

 support the simplifying assumption. 



We should be open to the possibility that the reason we have been waiting so 

 long for the mathematical apparatus which can render explicit useful informa- 

 tion implicitly contained in the Navier -Stokes equations, is that the development 



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